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Jonathan Borwein FRSC FAA FAAS www.carma.newcastle.edu.au/~jb616Laureate Professor University of Newcastle, NSW Director, Centre for Computer Assisted Research Mathematics and Applications CARMA

Revised 16-06-10: joint work with B. Sims. Thanks also to Ulli Kortenkamp and Chris Maitland

Clemson REU version ofANZIAM 2010 Lecture Queenstown, NZ: Jan 31st to Feb 4th 2010 Douglas-Ratchford iterations in the absence of convexity

Charles Darwins notes

Alan Turings Enigma

1THE REST IS SOFTWARE

It was my luck (perhaps my bad luck) to be the world chess champion during the critical years in which computers challenged, then surpassed, human chess players.Before 1994 and after 2004 these duels held little interest. - Garry Kasparov, 2010 http://www.carma.newcastle.edu.au/sigmaopt

2SIGMAOPT

3ABSTRACTThe Douglas-Rachford iteration scheme, introduced half a century ago in connection with nonlinear heat flow problems, aims to find a point common to two or more closed constraint sets.Convergence is ensured when the sets are convex subsets of a Hilbert space, however, despite the absence of satisfactory theoretical justification, the scheme has been routinely used to successfully solve a diversity of practical optimization or feasibility problems in which one or more of the constraints involved is non-convex. As a first step toward addressing this deficiency, we provide convergence results for a proto-typical non-convex scenario.

4PHASE RECONSTRUCTIONxPA(x)RA(x)A

2007 Elser solving Sudoku with reflectors

A2008 Finding exoplanet Fomalhaut in Piscis with projectors Projectors and Reflectors: PA(x) is the metric projection or nearest point and RA(x) reflects in the tangent: x is red "All physicists and a good many quite respectable mathematicians are contemptuous about proof." G. H. Hardy (1877-1947)

The story of Hubbles 1.3mm error in the upside down lens (1990)

And Keplers hunt for exo-planets (launched March 2009)6

WHY DOES IT WORK?Consider the simplest case of a line A of height h and the unit circle B. With the iteration becomes

In a wide variety of hard problems (protein folding, 3SAT, Sudoku) B is non-convex but DR and divide and concur (below) works better than theory can explain. It is:

An ideal problem for introducing early under-graduates to research, with many many accessible extensions in 2 or 3 dimensions For h=0 We prove convergence to one of the two points in A B iff we do not start on the vertical axis (where we have chaos). For h>1 (infeasible) it is easy to see the iterates go to infinity (vertically). For h=1 we converge to an infeasible point. For h in (0,1) the pictures are lovely but proofs escaped us for 9 months. Two representative Maple pictures follow:INTERACTIVE PHASE RECOVERY in CINDERELLARecall the simplest case of a line A of height h and the unit circle B. With the iteration becomes

A Cinderella picture of two steps from (4.2,-0.51) follows: The link is to reflection.htmlDIVIDE AND CONCUR

Serial (L) and Parallel (R)9CAS+IGP: THE GRIEF IS IN THE GUI

Numerical errors in using double precision

Accuracy after taking input from MapleLink is expansion.cdy Exploration first in Maple and then in Cinderella- ability to look at orbits/iterations dynamically is great for insight allows for rapid reinforcement and elaboration of intuition

Decided to look at ODE analogues- and their linearizations hoped for Lyapunov like results

Searched literature for a discrete version- found Perrons workTHE ROUTE TO DISCOVERY

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THE BASIS OF THE PROOF

Explains spin for height in (0,1)12WHAT WE CAN NOW SHOW

13COMMENTS and OPEN QUESTIONS As noted, the methods parallelize very well.

Explain why alternating projections works so well for optical aberration but not for phase reconstruction?

Show global convergence in the appropriate basins?

Extend to more general pairs of sets (and metrics)even half-line is much more complex try a sphere or a line and n collinear points

14REFERENCES[1] Jonathan M. Borwein and Brailey Sims, The Douglas-Rachford algorithm in the absence of convexity. Submitted,Fixed-Point Algorithms for Inverse Problems in Science and Engineering, to appear inSpringer Optimization and Its Applications, Nov. 2009.http://www.carma.newcastle.edu.au/~jb616/dr.pdf

[2] J. M. Borwein and J. Vanderwerff,Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and Applications, 109 Cambridge Univ Press, 2010.http://projects.cs.dal.ca/ddrive/ConvexFunctions/

[3] V. Elser, I. Rankenburg, and P. Thibault, Searching with iterated maps, Proc. National Academy of Sciences, 104 (2007), 418-423.15